Suppose I have the following stationary $AR(1)$ process:
$$ y_{t}=\alpha+\rho y_{t-1} + u_{t} $$
where $u_{t} \sim \mathbb{N}(0,\sigma^{2})$. Suppose I have an initial condition $y_{0}$ and terminal condition $y_{T}$ and I would like to simulate my process for the periods in the interim, i.e $t = 2,\dots,T-1$. Can someone tell me what is the right distribution from which I should draw the $u_{t}$?